Figure 1

In the Fun Fact on Spherical Geometry, we saw
an example of a space which is curved in such a way that
the sum of angles in a triangle is greater than 180 degrees,
where the sides of the triangle are "intrinsically" straight
lines, or geodesics.
Is it also possible to have a space that "curves" in such
a way that the sum of angles in a triangle is less than
180 degrees?
Yes! For instance, consider a saddleshaped surface.
A triangle that extends over the saddle of this surface
(whose edges are geodesics) will have this property.
Another space with this property is
something called the hyperbolic plane.
This can be modeled by
disc in which is "curved" in such a strange
way that a bug on this disc would think that
the "straight" lines are the
pieces of circles or straight lines
(viewed in planar geometry)
that intersect the disc boundary at right angles.
Any 3sided figure using such lines will
have angles in the corners that sum to less than
180 degrees!
Presentation Suggestions:
Convince students of the triangle assertion
by drawing a saddleshaped surface and a triangle on it.
Alternatively, you could
show that the angles of a square do not add to 360 degrees.
Follow by showing drawing the hyperbolic disc and
explaining what the "straight lines" are. You can
also construct and bring to class an
approximate physical model of a hyperbolic plane;
the references discuss ways to construct them.
The Math Behind the Fact:
These spaces are examples of spaces
with a kind of nonEuclidean geometry called
hyperbolic geometry.
Unlike planar geometry, the parallel postulate
does not hold in hyperbolic geometry. Two lines are said
to be parallel if they do not intersect. In
Euclidean geometry, given a line L there is exactly one
line through any given point P that is parallel to L
(the parallel postulate). However in hyperbolic
geometry, there are infinitely many lines parallel to
L passing through P.
Mathematicians sometimes work with strange geometries by
defining them in terms of a Riemannian metric, which
gives a local notion of how to measure "distance" and
"angles" on an arbitrary set.
You can learn more about such metrics by taking
a first course on real analysis, then following with an
advanced course in differential geometry.
How to Cite this Page:
Su, Francis E., et al. "Hyperbolic Geometry."
Math Fun Facts.
<http://www.math.hmc.edu/funfacts>.
